Pavel Stránský

29th August 2011

WHAT DRIVES NUCLEI TO BE PROLATE?

Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México

Alejandro FrankRoelof Bijker

CGS14, University of Guelph, Ontario, Canada, 2011

Experimental deformation of nuclei

N.J. Stone, At. Data Nucl. Data Tables 90, 75 (2005)

rare-earth region

is a typical value for well-deformed nuclei

Deformation parameter (from measured quadrupole moments):

where measured intrinsic

WHAT DRIVES NUCLEI TO BE PROLATE?

Surface tensionCoulomb energy…

Shell structureSpin-orbit and l2 interaction…

Macroscopic effects: Microscopic effects:

?

Minimization of the total sum of the lowest-lying occupied one-particle energies with respect to the size of the potential deformation

Minimization of the equilibrium energy with respect to the size of the shape deformation

Stable ground-state configuration

Microscopic single-particle models

1. Single-particle models

(a short discussion)

3D spheroid potential (axially symmetric elipsoid)

Pure harmonic potential

Equal number of prolate and oblate

configurations

Infinite potential well

N

• Volume saturation of the nuclear force• Sharp surface

V = const

1. Single-particle models

Noninteracting fermions (only 1 type of particles)

N

1s

1p

1d2s

2p

1f

1g

2d1h 2s

012

3

4

01

23

4

Level dynamics – Spheroid infinite well

E (a.u.)

Projection of the

angular momentum

1. Single-particle models

I.Hamamoto, B.R. Mottelson, Phys. Rev. C 79, 034317 (2009)

Sharp surface pushes down shells with higher orbital momentum l, containing additional downsloping states with low projection m on the prolate side; the predominance of these low-m states, together with their mutual repulsion, causes the prolate-oblate deformation asymmetry

Deformed liquid drop model(A little of the theory and results)

Total mass/energy (Weizsäcker formula)

volume energy surface energy Coulomb energy

A = N + Z

Adjustable constants:

Shape functions:

binding (bulk) energy

microscopic corrections

(asymmetry energy, shell effects, pairing)

curvature energy, surface and volume redistribution energy…

2. Deformed liquid drop model

Quadrupole deformation

Fixed by a condition of volume

conservation

2 < 0

2 = 0

2 > 0(axially symmetric)

oblate prolate

spherical

Deformation parameter

Symmetric with respect to the sign of 2

Negative for 2 < 0 – prolate shape has always lower energy

Surface

Coulomb

shape functions:

Numerically

W.J. Swiatecki, Phys. Rev. 104, 993 (1956)

2. Deformed liquid drop model

Prolate-oblate energy difference

keV

2. Deformed liquid drop model

rare-earth region

keV

Prolate-oblate energy difference

2. Deformed liquid drop model

surface

Coulomb

surface

Coulomb

Almost the same contribution (despite the different functional form)

Coulomb and surface contribution

2. Deformed liquid drop model

B from the B(E2) transition probabilities

S. Raman, C.W. Nestor, and P. Tikkanen, At. Data Nucl. Data Tables 78, 1 (2001)

- Only absolute value of the deformation- Only even-even nuclei

2. Deformed liquid drop model

Distribution of B values

495 nuclei totally

2. Deformed liquid drop model

Shortcomings of the pure LDM

Shape stabilizationPure liquid drop model is not able

to explain the ground state deformation (spherical shape is

always preferred)

Necessity of introducing shell correctionsShell corrections (Strutinsky)

N

E

Exact cumulative level density

Smooth cumulative level

densityspherical

deformed deformation decreases the size of the corrections

2. Deformed liquid drop model

Necessity of introducing shell corrections

Pure liquid drop model is not able to explain the ground state

deformation (spherical shape is always preferred)

Shape stabilization

2. Deformed liquid drop model

Symmetric with respect to the sign of the deformation

W.D. Myers, W.J. Swiatecki, Nucl. Phys. 81, 1 (1966)

Necessity of introducing shell corrections

Pure liquid drop model is not able to explain the ground state

deformation (spherical shape is always preferred)

Shape stabilization

Shell effects (1st approximation)

2. Deformed liquid drop model

Size of the shell corrections

40 80 120

Mid-shell correction < 3MeV

Shell corrections are highly important near closed shells, but less for deformed nuclei in mid-shells

S (

N,Z

)

Negative corrections:deepen the spherical minimum

Positive corrections:Create the oblate and prolate

minima

2. Deformed liquid drop model

Conclusions & Outlook• Collective effects (surface and Coulomb energy of the quadrupole

deformed simple liquid drop model) give a significant amount of the prolate-oblate energy difference up to B = 800keV (for comparison, the first 2+ excited state for well-deformed even-even nuclei is typically of the order of 100keV)

• This model is not capable of explaining the origin of the deformation: In order to stabilize a deformed shape, microscopic corrections (that may lower the prolate minimum, however) must be included

• Microscopic pure single-particle models explain the prolate preponderance as a consequence of the sharp surface and saturation of the nuclear matter. Complex calculations (such as the self-consistent the HF+BCS or the shell model with random interactions) favor the prolate shape, but the underlying responsible physics is hidden

• In the future: To find a link between the microscopic shell structure (i.g. the ordering of levels) and the exact shape of a nucleus

Last slide

Thank you very much for your

attention